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At the critical temperature (Tc ≈ 2.27), the system undergoes a phase transition. Look for large "fractal-like" clusters of spins forming and dissolving.
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Magnetization History
The Mathematics of Magnetism
The Ising model simplifies a magnet into a grid of atomic "spins" that can point either up (+1) or down (-1). Atoms want to align with their neighbors (ferromagnetism) and with any external magnetic field.
// Hamiltonian (Total Energy)
H(σ) = -J ∑ σiσj - B ∑ σi
The Phase Transition
At high temperatures, thermal noise flips spins randomly, destroying order (Paramagnetic phase). At low temperatures, neighbor interactions win, creating large domains of aligned spins (Ferromagnetic phase). The sharp change occurs at the Critical Temperature (Tc).
Iron filings revealing magnetic field lines.
Illustration of early 20th century theoretical physics.
A Thesis That Became a Legend
In 1920, physicist Wilhelm Lenz proposed a model of ferromagnetism to his student, Ernst Ising. Ising solved the model for a 1-dimensional chain in his 1924 PhD thesis link.
Ising found no phase transition in 1D (the chain stays disordered at any T > 0). He incorrectly assumed this held for 3D as well. The model languished until 1936, when Rudolf Peierls argued that 2D and 3D versions could have phase transitions.
The Onsager Solution (1944)
The exact solution for the 2D case by Lars Onsager is considered one of the tour-de-force achievements of mathematical physics, proving definitively that simple interaction rules can generate complex phase transitions link.
Today, the Ising model is arguably the most studied model in statistical physics, serving as the "fruit fly" for understanding complexity.
Neuroscience & Hopfield Nets
In the brain, neurons firing (1) or resting (-1) can be modeled like spins. Hopfield Networks are a form of Ising model where "energy minima" correspond to stored memories. The system "relaxes" into a memory state, similar to how a magnet relaxes into alignment.
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Social Dynamics & Voters
In Voter Models, an individual's opinion (Left/Right) is influenced by their neighbors. High "social temperature" (noise/misinformation) prevents consensus, while strong peer pressure (high J) leads to polarization (domains), exactly like magnetic domains link.
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